Apaper by Soheil Kolouri and co-authors was arXived last week about using Wasserstein distance for inference on multivariate Gaussian mixtures. The basic concept is that the parameter is estimated by minimising the p-Wasserstein distance to the empirical distribution, smoothed by a Normal kernel. As the general Wasserstein distance is quite costly to compute, the approach relies on a sliced version, which means computing the Wasserstein distance between one-dimensional projections of the distributions. Optimising over the directions is an additional computational constraint.

“To fit a finite GMM to the observed data, one is required to answer the following questions: 1) how to estimate the number of mixture components needed to represent the data, and 2) how to estimate the parameters of the mixture components.”

The paper contains a most puzzling comment opposing maximum likelihood estimation to minimum Wasserstein distance estimation on the basis that the later would not suffer from multimodality. This sounds incorrect as the multimodality of a mixture model (likelihood) stems from the lack of identifiability of the parameters. If all permutations of these parameters induce exactly the same distribution, they all stand at the same distance from the data distribution, whatever the distance is. Furthermore, the above tartan-like picture clashes with the representation of the log-likelihood of a Normal mixture, as exemplified by the picture below based on a 150 sample with means 0 and 2, same unit variance, and weights 0.3 and 0.7, which shows a smooth if bimodal structure:And for the same dataset, my attempt at producing a Wasserstein “energy landscape” does return a multimodal structure (this is the surface of minus the logarithm of the 2-Wasserstein distance):“Jin et al. proved that with random initialization, the EM algorithm will converge to a bad critical point with high probability.”

This statement is most curious in that the “probability” in the assessment must depend on the choice of the random initialisation, hence on a sort of prior distribution that is not explicited in the paper. Which remains blissfully unaware of Bayesian approaches.

Another [minor mode] puzzling statement is that the p-Wasserstein distance is defined on the space of probability measures with finite p-th moment, which does not make much sense when what matters is rather the finiteness of the expectation of the distance d(X,Y) raised to the power p. A lot of the maths details either do not make sense or seem superfluous.

“I have decided that mixtures, like tequila, are inherently evil and should be avoided at all costs.” L. Wasserman

Larry Wasserman once remarked that finite mixtures were like the twilight zone of statistics, thanks to the numerous idiosyncrasies associated with such models. And George Casella had similar strong reservations about mixture estimation. Avi Feller and co-authors [including Natesh Pillai] have just arXived a paper on this topic, exhibiting shocking (!) properties of the MLE! Their core example is a mixture of two normal distributions with known common variance and known weight different from 0.5, which ensures identifiability. This is a favourite example of mine that we used for instance in our book Introducing Monte Carlo methods with R. If only because we can plot the likelihood and posterior surfaces. (Warning: I wrote those notes on an earlier version of the paper, so mileage may vary in terms of accuracy!)

The “shocking” discovery in the paper is that the MLE is wrong as often as not in selecting the sign of the difference Δ between both means, with an additional accumulation point at zero. The global mode may thus be in the wrong place for small enough sample sizes. And even for larger sizes: when the difference between the means is small the likelihood is likely to be unimodal with a mode quite close to zero. (An interesting remark is that the likelihood derivative is always zero at Δ=0 when considering the special case of both means equal to -Δ and to πΔ/(1-π), respectively, which implies that the overall mean of the mixture is equal to zero. A potential connection with our reparameterisation paper, maybe?)

The alternative proposed by Avi and his co-authors is to proceed through moments, i.e., to revert to Pearson (1892). There are however difficulties with this approach, first and foremost the non-uniqueness of the moment equations used to estimate Δ. For instance, the second cumulant equation chosen by the authors is not always defined as opposed to the third cumulant equation (why not using this third cumulant then). Which does not always produce the right sign… But, in a strange twist, the authors turn those deficiencies into signals for both pathologies (wrong sign and “pile-up” at zero).

The most importance issue in this framework being in estimating the parameters, the authors opt for an approach based on tests, which is definitely surprising given the well-known deficiencies of standard tests in mixtures. The test chosen here is a Wald test with a statistic equal to the χ² version of the first cumulant differences. I am surprised that the χ² approximation works in such an unfriendly setting. And I do not understand how the grid is used, unless a certain degree of approximation is accepted, which takes us back to the “dark ages” of imposing a minimal distance Δ to achieve consistency, as in Ghosh and Sen (1985).

“..our concern about sign error is trivial in the Bayesian setting: the global mode is simply a poor summary of a multi-modal posterior. More broadly, the weak identification issues we highlight in this paper are not necessarily relevant to a strict Bayesian.”

A priori, I do not think pathologies of the MLE always transfer to Bayes estimators, unless one uses the MAP as an [poor] estimator. But using the MAP is not necessary since posterior means are meaningful in this identified setting, where label switching should not occur. However, running the same experiments with a Gaussian prior on both means and using the posterior mean as my estimator, I did obtain the same pathology of Bayes estimates [also produced in the supplementary material] not concentrating on the true value of the difference, but putting weight on the opposite value and at zero. Using a less standard prior inspired by David Rossell’s talk on non-local priors two weeks ago, which avoids a neighbourhood of zero, I did not get a much different picture as illustrated below:

Overall, I remain somewhat uncertain as to what to conclude from this pathological behaviour. When both means are close enough, the sign of the difference is often estimated wrongly. But that could simply mean that the means are not significantly different, for that sample size…

Hyungsuk Tak, Xiao-Li Meng and David van Dyk just arXived a paper on a multiple choice proposal in Metropolis-Hastings algorithms towards dealing with multimodal targets. Called “A repulsive-attractive Metropolis algorithm for multimodality” [although I wonder why XXL did not jump at the opportunity to use the “love-hate” denomination!]. The proposal distribution includes a [forced] downward Metropolis-Hastings move that uses the inverse of the target density π as its own target, namely 1/{π(x)+ε}. Followed by a [forced] Metropolis-Hastings upward move which target is {π(x)+ε}. The +ε is just there to avoid handling ratios of zeroes (although I wonder why using the convention 0/0=1 would not work). And chosen as 10⁻³²³ by default in connection with R smallest positive number. Whether or not the “downward” move is truly downwards and the “upward” move is truly upwards obviously depends on the generating distribution: I find it rather surprising that the authors consider the same random walk density in both cases as I would have imagined relying on a more dispersed distribution for the downward move in order to reach more easily other modes. For instance, the downward move could have been based on an anti-Langevin proposal, relying on the gradient to proceed further down…

This special choice of a single proposal however simplifies the acceptance ratio (and keeps the overall proposal symmetric). The final acceptance ratio still requires a ratio of intractable normalising constants that the authors bypass by Møller et al. (2006) auxiliary variable trick. While the authors mention the alternative pseudo-marginal approach of Andrieu and Roberts (2009), they do not try to implement it, although this would be straightforward here: since the normalising constants are the probabilities of accepting a downward and an upward move, respectively. Those can easily be evaluated at a cost similar to the use of the auxiliary variables. That is,

– generate a few moves from the current value and record the proportion p of accepted downward moves;
– generate a few moves from the final proposed value and record the proportion q of accepted downward moves;

and replace the ratio of intractable normalising constants with p/q. It is not even clear that one needs those extra moves since the algorithm requires an acceptance in the downward and upward moves, hence generate Geometric variates associated with those probabilities p and q, variates that can be used for estimating them. From a theoretical perspective, I also wonder if forcing the downward and upward moves truly leads to an improved convergence speed. Considering the case when the random walk is poorly calibrated for either the downward or upward move, the number of failed attempts before an acceptance may get beyond the reasonable.

As XXL and David pointed out to me, the unusual aspect of the approach is that here the proposal density is intractable, rather than the target density itself. This makes using Andrieu and Roberts (2009) seemingly less straightforward. However, as I was reminded this afternoon at the statistics and probability seminar in Bristol, the argument for the pseudo-marginal based on an unbiased estimator is that w Q(w|x) has a marginal in x equal to π(x) when the expectation of w is π(x). In thecurrent problem, the proposal in x can extended into a proposal in (x,w), w P(w|x) whose marginal is the proposal on x.

If we complement the target π(x) with the conditional P(w|x), the acceptance probability would then involve

Ten days ago, Gersende Fort, Benjamin Jourdain, Tony Lelièvre, and Gabriel Stoltz arXived a study about an adaptive umbrella sampler that can be re-interpreted as a Wang-Landau algorithm, if not the most efficient version of the latter. This reminded me very much of the workshop we had all together in Edinburgh last June. And even more of the focus of the molecular dynamics talks in this same ICMS workshop about accelerating the MCMC exploration of multimodal targets. The self-healing aspect of the sampler is to adapt to the multimodal structure thanks to a partition that defines a biased sampling scheme spending time in each set of the partition in a frequency proportional to weights. While the optimal weights are the weights of the sets against the target distribution (are they truly optimal?! I would have thought lifting low density regions, i.e., marshes, could improve the mixing of the chain for a given proposal), those are unknown and they need to be estimated by an adaptive scheme that makes staying in a given set the less desirable the more one has visited it. By increasing the inverse weight of a given set by a factor each time it is visited. Which sounds indeed like Wang-Landau. The plus side of the self-healing umbrella sampler is that it only depends on a scale γ (and on the partition). Besides converging to the right weights of course. The downside is that it does not reach the most efficient convergence, since the adaptivity weight decreases in 1/n rather than 1/√n.

Note that the paper contains a massive experimental side where the authors checked the impact of various parameters by Monte Carlo studies of estimators involving more than a billion iterations. Apparently repeated a large number of times.

The next step in adaptivity should be about the adaptive determination of the partition, hoping for a robustness against the dimension of the space. Which may be unreachable if I judge by the apparent deceleration of the method when the number of terms in the partition increases.

Label switching is an issue with mixture estimation (and other latent variable models) because mixture models are ill-posed models where part of the parameter is not identifiable. Indeed, the density of a mixture being a sum of terms

the parameter (vector) of the ω’s and of the θ’s is at best identifiable up to an arbitrary permutation of the components of the above sum. In other words, “component #1 of the mixture” is not a meaningful concept. And hence cannot be estimated.

This problem has been known for quite a while, much prior to EM and MCMC algorithms for mixtures, but it is only since mixtures have become truly estimable by Bayesian approaches that the debate has grown on this issue. In the very early days, Jean Diebolt and I proposed ordering the components in a unique way to give them a meaning. For instant, “component #1” would then be the component with the smallest mean or the smallest weight and so on… Later, in one of my favourite X papers, with Gilles Celeux and Merrilee Hurn, we exposed the convergence issues related with the non-identifiability of mixture models, namely that the posterior distributions were almost always multimodal, with a multiple of k! symmetric modes in the case of exchangeable priors, and therefore that Markov chains would have trouble to visit all those modes in a symmetric manner, despite the symmetry being guaranteed from the shape of the posterior. And we conclude with the slightly provocative statement that hardly any Markov chain inferring about mixture models had ever converged! In parallel, time-wise, Matthew Stephens had completed a thesis at Oxford on the same topic and proposed solutions for relabelling MCMC simulations in order to identify a single mode and hence produce meaningful estimators. Giving another meaning to the notion of “component #1”.

And then the topic began to attract more and more researchers, being both simple to describe and frustrating in its lack of definitive answer, both from simulation and inference perspectives. Rodriguez’s and Walker’s paper provides a survey on the label switching strategies in the Bayesian processing of mixtures, but its innovative part is in deriving a relabelling strategy. Which consists of finding the optimal permutation (at each iteration of the Markov chain) by minimising a loss function inspired from k-means clustering. Which is connected with both Stephens’ and our [JASA, 2000] loss functions. The performances of this new version are shown to be roughly comparable with those of other relabelling strategies, in the case of Gaussian mixtures. (Making me wonder if the choice of the loss function is not favourable to Gaussian mixtures.) And somehow faster than Stephens’ Kullback-Leibler loss approach.

“Hence, in an MCMC algorithm, the indices of the parameters can permute multiple times between iterations. As a result, we cannot identify the hidden groups that make [all] ergodic averages to estimate characteristics of the components useless.”

One section of the paper puzzles me, albeit it does not impact the methodology and the conclusions. In Section 2.1 (p.27), the authors consider the quantity

which is the marginal probability of allocating observation i to cluster or component j. Under an exchangeable prior, this quantity is uniformly equal to 1/k for all observations i and all components j, by virtue of the invariance under permutation of the indices… So at best this can serve as a control variate. Later in Section 2.2 (p.28), the above sentence does signal a problem with those averages but it seem to attribute it to MCMC behaviour rather than to the invariance of the posterior (or to the non-identifiability of the components per se). At last, the paper mentions that “given the allocations, the likelihood is invariant under permutations of the parameters and the allocations” (p.28), which is not correct, since eqn. (8)

does not hold when the two permutations σ and τ give different images of zi…

Last and maybe most exciting day of the “High-dimensional Stochastic Simulation and Optimisation in Image Processing” in Bristol as it was exclusively about simulation (MCMC) methods. Except my own talk on ABC. And Peter Green’s on consistency of Bayesian inference in non-regular models. The talks today were indeed about using convex optimisation devices to speed up MCMC algorithms with tools that were entirely new to me, like the Moreau transform discussed by Marcelo Pereyra. Or using auxiliary variables à la RJMCMC to bypass expensive Choleski decompositions. Or optimisation steps from one dual space to the original space for the same reason. Or using pseudo-gradients on partly differentiable functions in the talk by Sylvain Lecorff on a paper commented earlier in the ‘Og. I particularly liked the notion of Moreau regularisation that leads to more efficient Langevin algorithms when the target is not regular enough. Actually, the discretised diffusion itself may be geometrically ergodic without the corrective step of the Metropolis-Hastings acceptance. This obviously begs the question of an extension to Hamiltonian Monte Carlo. And to multimodal targets, possibly requiring as many normalisation factors as there are modes. So, in fine, a highly informative workshop, with the perfect size and the perfect crowd (which happened to be predominantly French, albeit from a community I did not have the opportunity to practice previously). Massive kudos to Marcello for putting this workshop together, esp. on a week where family major happy events should have kept him at home!

As the workshop ended up in mid-afternoon, I had plenty of time for a long run with Florence Forbes down to the Avon river and back up among the deers of Ashton Court, avoiding most of the rain, all of the mountain bikes on a bike trail that sounded like trail running practice, and building enough of an appetite for the South Indian cooking of the nearby Thali Café. Brilliant!